A) \[\left[ 0,\,\,\frac{\pi }{6} \right)\]
B) \[\left[ \frac{5\pi }{6},\,\,\pi \right]\]
C) \[\left[ \frac{\pi }{6},\,\,\frac{\pi }{2} \right]\]
D) \[\left[ \frac{\pi }{2},\,\,\frac{5\pi }{6} \right]\]
Correct Answer: A
Solution :
We have,\[|\mathbf{a}-\mathbf{b}{{|}^{2}}=|\mathbf{a}{{|}^{2}}+|\mathbf{b}{{|}^{2}}-2(\mathbf{a}\cdot \mathbf{b})\] \[\Rightarrow \]\[|\mathbf{a}-\mathbf{b}{{|}^{2}}=|\mathbf{a}{{|}^{2}}+|\mathbf{b}{{|}^{2}}-2|\mathbf{a}||\mathbf{b}{{|}^{2}}\cos 2\theta \] \[\Rightarrow \]\[|\mathbf{a}-\mathbf{b}{{|}^{2}}=2-2\cos 2\theta \]\[[\because |\mathbf{a}|=|\mathbf{b}|=1]\] \[\Rightarrow \]\[|\mathbf{a}-\mathbf{b}{{|}^{2}}=4{{\sin }^{2}}\theta \] \[\Rightarrow \]\[|\mathbf{a}-\mathbf{b}|=2|\sin \theta |\] Now,\[|\mathbf{a}-\mathbf{b}|<1\Rightarrow 2|\sin \theta |<1\] \[\Rightarrow \] \[|\sin \theta |<\frac{1}{2}\] \[\Rightarrow \] \[\theta \in \left[ 0,\,\,\frac{\pi }{6} \right)or\left( \frac{5\pi }{6},\,\,\pi \right]\]
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